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Statistics 2 > Week 4
All PYQs
Topic Wise PYQs
Start Weekly Test
Type:
All
Difficulty:
All
Year:
All
36Q
01
Let
X
X
X
be a continuous random variable with the following CDF:
F
X
(
x
)
=
{
0
,
x
<
−
1
(
x
+
1
)
2
2
,
−
1
≤
x
<
0
1
−
(
1
−
x
)
2
2
,
0
≤
x
<
1
1
,
x
≥
1
F_X(x)=\begin{cases}0, & x<-1\\\frac{(x+1)^2}{2}, & -1\le x<0\\1-\frac{(1-x)^2}{2}, & 0\le x<1\\1, & x\ge 1\end{cases}
F
X
(
x
)
=
⎩
⎨
⎧
0
,
2
(
x
+
1
)
2
,
1
−
2
(
1
−
x
)
2
,
1
,
x
<
−
1
−
1
≤
x
<
0
0
≤
x
<
1
x
≥
1
Which of the following is the correct PDF of
X
X
X
?
Single correct
MEDIUM
3 marks
26 October 2025
02
Find the value of
k
k
k
.
Comprehension
MEDIUM
2 marks
26 October 2025
03
Find the value of
P
(
1
<
X
<
2
∣
X
>
1
2
)
P(1<X<2 \mid X>\frac{1}{2})
P
(
1
<
X
<
2
∣
X
>
2
1
)
. Enter the answer correct to two decimal places.
Comprehension
HARD
3 marks
26 October 2025
04
The time (in minutes) a customer spends inside a service center follows an exponential distribution with an average of
10
10
10
minutes. What is the probability that a customer has to wait at least
12
12
12
more minutes in the center given that he has already spent
10
10
10
minutes in the center? Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
13 July 2025
05
Calculate the value of
Var
(
X
)
\operatorname{Var}(X)
Var
(
X
)
.
Comprehension
MEDIUM
2 marks
13 July 2025
06
Define
Y
=
72
X
+
5
Y=72X+5
Y
=
72
X
+
5
, then find the value of
Var
(
Y
)
\operatorname{Var}(Y)
Var
(
Y
)
.
Comprehension
EASY
1 marks
13 July 2025
07
Find the value of
a
+
b
a+b
a
+
b
.
Comprehension
EASY
2 marks
13 July 2025
08
If the value of
b
b
b
is
14
14
14
, then find the value of
P
(
5
<
T
<
11
∣
T
>
7
)
P(5<T<11\mid T>7)
P
(
5
<
T
<
11
∣
T
>
7
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
13 July 2025
09
The probability density function of a random variable
X
X
X
is given as
f
X
(
x
)
=
{
c
,
0
≤
x
<
1
4
2
c
,
1
4
≤
x
<
3
4
3
c
,
3
4
≤
x
<
1
0
,
otherwise
f_X(x)=\begin{cases}c, & 0\le x<\frac{1}{4}\\2c, & \frac{1}{4}\le x<\frac{3}{4}\\3c, & \frac{3}{4}\le x<1\\0, & \text{otherwise}\end{cases}
f
X
(
x
)
=
⎩
⎨
⎧
c
,
2
c
,
3
c
,
0
,
0
≤
x
<
4
1
4
1
≤
x
<
4
3
4
3
≤
x
<
1
otherwise
Find the value of
c
c
c
. Enter the answer correct to one decimal place.
Numerical
MEDIUM
3 marks
23 February 2025
10
Find the value of
a
a
a
.
Comprehension
EASY
1 marks
23 February 2025
11
What is the CDF of
X
X
X
?
Comprehension
MEDIUM
2 marks
23 February 2025
12
What is the value of
P
(
−
4
<
X
≤
6
)
P(-4<X\le6)
P
(
−
4
<
X
≤
6
)
?
Comprehension
MEDIUM
2 marks
23 February 2025
13
A tech company develops a new software system. The time until the system experiences its first major failure is modeled by an exponential distribution with an average failure time of
5
5
5
years. Find the value of
k
k
k
such that there is a
95
%
95\%
95%
chance the system will fail within
k
k
k
years.
Single correct
MEDIUM
3 marks
27 October 2024
14
Find the value of
k
k
k
. Enter the answer correct to two decimal places.
Comprehension
EASY
2 marks
27 October 2024
15
Calculate the probability that the length of a randomly selected rod is between
2
2
2
cm and
4
4
4
cm. Enter the answer correct to two decimal places.
Comprehension
EASY
3 marks
27 October 2024
16
Suppose that the number of miles that a car can run before its battery wears out is exponentially distributed, with an average value of
10000
10000
10000
miles. If you desire to take a
5000
5000
5000
-mile trip, what is the probability that you will be able to complete the trip without replacing the car battery?
Single correct
MEDIUM
3 marks
07 July 2024
17
A teacher observes that the cumulative distribution function (CDF) for the scores on a mathematics test is
F
(
x
)
=
{
0
,
x
<
0
,
x
2
,
0
≤
x
≤
1
,
1
,
x
>
1.
F(x)=\begin{cases}0, & x<0,\\ x^2, & 0\le x\le1,\\ 1, & x>1.\end{cases}
F
(
x
)
=
⎩
⎨
⎧
0
,
x
2
,
1
,
x
<
0
,
0
≤
x
≤
1
,
x
>
1.
For
0
≤
x
≤
1
0\le x\le1
0
≤
x
≤
1
, where
x
x
x
is the score normalized to
1
1
1
. If a student scores above
0.7
0.7
0.7
, what is the conditional probability that they actually score above
0.9
0.9
0.9
? Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
07 July 2024
18
Find the value of
c
c
c
so that
f
X
(
x
)
f_X(x)
f
X
(
x
)
is a valid PDF.
Comprehension
MEDIUM
2 marks
07 July 2024
19
Calculate the CDF of
X
X
X
.
Comprehension
MEDIUM
3 marks
07 July 2024
20
The CDF of a random variable
X
X
X
is given as
F
X
(
x
)
=
1
−
e
−
4
x
F_X(x)=1-e^{-4x}
F
X
(
x
)
=
1
−
e
−
4
x
for
x
≥
0
x\ge0
x
≥
0
, and
0
0
0
otherwise. What is the value of
P
(
−
4
<
X
≤
6
)
P(-4<X\le6)
P
(
−
4
<
X
≤
6
)
?
Single correct
EASY
3 marks
25 February 2024
21
Find the value of
k
k
k
. Enter the answer correct to one decimal place.
Comprehension
EASY
2 marks
25 February 2024
22
What is the value of
P
(
1
<
X
<
2.5
)
P(1<X<2.5)
P
(
1
<
X
<
2.5
)
? Enter the answer correct to two decimal places.
Comprehension
EASY
3 marks
25 February 2024
23
Consider a function
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
f
(
x
)
=
c
x
2
f(x)=cx^2
f
(
x
)
=
c
x
2
for
1
≤
x
<
2
1\le x<2
1
≤
x
<
2
,
f
(
x
)
=
c
x
f(x)=cx
f
(
x
)
=
c
x
for
2
≤
x
<
4
2\le x<4
2
≤
x
<
4
, and
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
otherwise, where
c
c
c
is any real constant. Find
c
c
c
such that
f
f
f
is a valid density function. Enter the answer correct to two decimal places. Hint:
∫
x
k
d
x
=
x
k
+
1
k
+
1
\int x^k\,dx=\frac{x^{k+1}}{k+1}
∫
x
k
d
x
=
k
+
1
x
k
+
1
.
Numerical
MEDIUM
3 marks
29 October 2023
24
Which of the following is true?
Comprehension
MEDIUM
2 marks
29 October 2023
25
Find
P
(
∣
X
∣
<
α
2
)
P(|X|<\frac{\alpha}{2})
P
(
∣
X
∣
<
2
α
)
. Enter the answer correct to two decimal places.
Comprehension
HARD
3 marks
29 October 2023
26
Let
X
X
X
be a continuous random variable with the following PDF:
f
X
(
x
)
=
k
(
1
+
x
)
2
f_X(x)=\frac{k}{(1+x)^2}
f
X
(
x
)
=
(
1
+
x
)
2
k
for
0
≤
x
≤
4
0\le x\le 4
0
≤
x
≤
4
, and
0
0
0
otherwise. Find the value of
k
k
k
. Enter the answer correct to two decimal places. Hint:
∫
1
(
a
+
b
x
)
2
d
x
=
−
1
b
(
a
+
b
x
)
\int\frac{1}{(a+bx)^2}\,dx=-\frac{1}{b(a+bx)}
∫
(
a
+
b
x
)
2
1
d
x
=
−
b
(
a
+
b
x
)
1
.
Numerical
MEDIUM
3 marks
16 July 2023
27
Find the value of
k
k
k
.
Comprehension
EASY
2 marks
16 July 2023
28
Find the value of
P
(
1
4
≤
X
≤
3
4
)
P(\frac{1}{4}\le X\le \frac{3}{4})
P
(
4
1
≤
X
≤
4
3
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
16 July 2023
29
Which of the following statements are correct?
Single correct
MEDIUM
3 marks
16 October 2022
30
Among the options below, for what values of
a
a
a
and
b
b
b
is the function
f
f
f
a valid density function?
Comprehension
MEDIUM
3 marks
16 October 2022
31
With the choice of
a
a
a
and
b
b
b
given in the previous question, find
P
(
X
≤
−
1
2
∣
X
<
1
)
P\left(X\le -\frac{1}{2}\mid X<1\right)
P
(
X
≤
−
2
1
∣
X
<
1
)
. Enter the answer correct to three decimal places.
Comprehension
MEDIUM
2 marks
16 October 2022
32
Let
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
be defined by
f
(
x
)
=
{
0
,
−
∞
<
x
<
0
,
1
4
,
0
<
x
≤
1
,
3
4
x
2
,
1
<
x
<
∞
.
f(x)=\begin{cases}0, & -\infty < x < 0,\\ \frac{1}{4}, & 0 < x \le 1,\\ \frac{3}{4x^2}, & 1 < x < \infty.\end{cases}
f
(
x
)
=
⎩
⎨
⎧
0
,
4
1
,
4
x
2
3
,
−
∞
<
x
<
0
,
0
<
x
≤
1
,
1
<
x
<
∞.
If a random variable
X
X
X
has the PDF
f
(
x
)
f(x)
f
(
x
)
, calculate
P
(
−
1
2
<
X
<
3
2
)
P\left(-\frac{1}{2}<X<\frac{3}{2}\right)
P
(
−
2
1
<
X
<
2
3
)
. Enter the answer correct to one decimal place. Hint: use
∫
a
b
1
x
2
d
x
=
1
a
−
1
b
\int_a^b \frac{1}{x^2}\,dx = \frac{1}{a}-\frac{1}{b}
∫
a
b
x
2
1
d
x
=
a
1
−
b
1
.
Numerical
MEDIUM
2 marks
05 June 2022
33
Find the value of
p
p
p
so that
f
X
(
x
)
f_X(x)
f
X
(
x
)
is a valid PDF. Enter the answer correct to two decimal places.
Numerical
EASY
2 marks
05 June 2022
34
Find the value of
P
(
X
<
4
∣
X
>
2
)
P(X<4\mid X>2)
P
(
X
<
4
∣
X
>
2
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
05 June 2022
35
What is the probability that Jadeja will bat for more than
40
40
40
minutes? Enter the answer correct to two decimal places.
Numerical
EASY
2 marks
05 June 2022
36
What is the probability that Jadeja will bat for more than
60
60
60
minutes given that he has already batted for more than
40
40
40
minutes? Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
05 June 2022
Showing 36 questions.
Statistics 2 > Week 4
All PYQs
Topic Wise PYQs
Start Weekly Test
Type:
All
Difficulty:
All
Year:
All
36Q
01
Let
X
X
X
be a continuous random variable with the following CDF:
F
X
(
x
)
=
{
0
,
x
<
−
1
(
x
+
1
)
2
2
,
−
1
≤
x
<
0
1
−
(
1
−
x
)
2
2
,
0
≤
x
<
1
1
,
x
≥
1
F_X(x)=\begin{cases}0, & x<-1\\\frac{(x+1)^2}{2}, & -1\le x<0\\1-\frac{(1-x)^2}{2}, & 0\le x<1\\1, & x\ge 1\end{cases}
F
X
(
x
)
=
⎩
⎨
⎧
0
,
2
(
x
+
1
)
2
,
1
−
2
(
1
−
x
)
2
,
1
,
x
<
−
1
−
1
≤
x
<
0
0
≤
x
<
1
x
≥
1
Which of the following is the correct PDF of
X
X
X
?
Single correct
MEDIUM
3 marks
26 October 2025
02
Find the value of
k
k
k
.
Comprehension
MEDIUM
2 marks
26 October 2025
03
Find the value of
P
(
1
<
X
<
2
∣
X
>
1
2
)
P(1<X<2 \mid X>\frac{1}{2})
P
(
1
<
X
<
2
∣
X
>
2
1
)
. Enter the answer correct to two decimal places.
Comprehension
HARD
3 marks
26 October 2025
04
The time (in minutes) a customer spends inside a service center follows an exponential distribution with an average of
10
10
10
minutes. What is the probability that a customer has to wait at least
12
12
12
more minutes in the center given that he has already spent
10
10
10
minutes in the center? Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
13 July 2025
05
Calculate the value of
Var
(
X
)
\operatorname{Var}(X)
Var
(
X
)
.
Comprehension
MEDIUM
2 marks
13 July 2025
06
Define
Y
=
72
X
+
5
Y=72X+5
Y
=
72
X
+
5
, then find the value of
Var
(
Y
)
\operatorname{Var}(Y)
Var
(
Y
)
.
Comprehension
EASY
1 marks
13 July 2025
07
Find the value of
a
+
b
a+b
a
+
b
.
Comprehension
EASY
2 marks
13 July 2025
08
If the value of
b
b
b
is
14
14
14
, then find the value of
P
(
5
<
T
<
11
∣
T
>
7
)
P(5<T<11\mid T>7)
P
(
5
<
T
<
11
∣
T
>
7
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
13 July 2025
09
The probability density function of a random variable
X
X
X
is given as
f
X
(
x
)
=
{
c
,
0
≤
x
<
1
4
2
c
,
1
4
≤
x
<
3
4
3
c
,
3
4
≤
x
<
1
0
,
otherwise
f_X(x)=\begin{cases}c, & 0\le x<\frac{1}{4}\\2c, & \frac{1}{4}\le x<\frac{3}{4}\\3c, & \frac{3}{4}\le x<1\\0, & \text{otherwise}\end{cases}
f
X
(
x
)
=
⎩
⎨
⎧
c
,
2
c
,
3
c
,
0
,
0
≤
x
<
4
1
4
1
≤
x
<
4
3
4
3
≤
x
<
1
otherwise
Find the value of
c
c
c
. Enter the answer correct to one decimal place.
Numerical
MEDIUM
3 marks
23 February 2025
10
Find the value of
a
a
a
.
Comprehension
EASY
1 marks
23 February 2025
11
What is the CDF of
X
X
X
?
Comprehension
MEDIUM
2 marks
23 February 2025
12
What is the value of
P
(
−
4
<
X
≤
6
)
P(-4<X\le6)
P
(
−
4
<
X
≤
6
)
?
Comprehension
MEDIUM
2 marks
23 February 2025
13
A tech company develops a new software system. The time until the system experiences its first major failure is modeled by an exponential distribution with an average failure time of
5
5
5
years. Find the value of
k
k
k
such that there is a
95
%
95\%
95%
chance the system will fail within
k
k
k
years.
Single correct
MEDIUM
3 marks
27 October 2024
14
Find the value of
k
k
k
. Enter the answer correct to two decimal places.
Comprehension
EASY
2 marks
27 October 2024
15
Calculate the probability that the length of a randomly selected rod is between
2
2
2
cm and
4
4
4
cm. Enter the answer correct to two decimal places.
Comprehension
EASY
3 marks
27 October 2024
16
Suppose that the number of miles that a car can run before its battery wears out is exponentially distributed, with an average value of
10000
10000
10000
miles. If you desire to take a
5000
5000
5000
-mile trip, what is the probability that you will be able to complete the trip without replacing the car battery?
Single correct
MEDIUM
3 marks
07 July 2024
17
A teacher observes that the cumulative distribution function (CDF) for the scores on a mathematics test is
F
(
x
)
=
{
0
,
x
<
0
,
x
2
,
0
≤
x
≤
1
,
1
,
x
>
1.
F(x)=\begin{cases}0, & x<0,\\ x^2, & 0\le x\le1,\\ 1, & x>1.\end{cases}
F
(
x
)
=
⎩
⎨
⎧
0
,
x
2
,
1
,
x
<
0
,
0
≤
x
≤
1
,
x
>
1.
For
0
≤
x
≤
1
0\le x\le1
0
≤
x
≤
1
, where
x
x
x
is the score normalized to
1
1
1
. If a student scores above
0.7
0.7
0.7
, what is the conditional probability that they actually score above
0.9
0.9
0.9
? Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
07 July 2024
18
Find the value of
c
c
c
so that
f
X
(
x
)
f_X(x)
f
X
(
x
)
is a valid PDF.
Comprehension
MEDIUM
2 marks
07 July 2024
19
Calculate the CDF of
X
X
X
.
Comprehension
MEDIUM
3 marks
07 July 2024
20
The CDF of a random variable
X
X
X
is given as
F
X
(
x
)
=
1
−
e
−
4
x
F_X(x)=1-e^{-4x}
F
X
(
x
)
=
1
−
e
−
4
x
for
x
≥
0
x\ge0
x
≥
0
, and
0
0
0
otherwise. What is the value of
P
(
−
4
<
X
≤
6
)
P(-4<X\le6)
P
(
−
4
<
X
≤
6
)
?
Single correct
EASY
3 marks
25 February 2024
21
Find the value of
k
k
k
. Enter the answer correct to one decimal place.
Comprehension
EASY
2 marks
25 February 2024
22
What is the value of
P
(
1
<
X
<
2.5
)
P(1<X<2.5)
P
(
1
<
X
<
2.5
)
? Enter the answer correct to two decimal places.
Comprehension
EASY
3 marks
25 February 2024
23
Consider a function
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
such that
f
(
x
)
=
c
x
2
f(x)=cx^2
f
(
x
)
=
c
x
2
for
1
≤
x
<
2
1\le x<2
1
≤
x
<
2
,
f
(
x
)
=
c
x
f(x)=cx
f
(
x
)
=
c
x
for
2
≤
x
<
4
2\le x<4
2
≤
x
<
4
, and
f
(
x
)
=
0
f(x)=0
f
(
x
)
=
0
otherwise, where
c
c
c
is any real constant. Find
c
c
c
such that
f
f
f
is a valid density function. Enter the answer correct to two decimal places. Hint:
∫
x
k
d
x
=
x
k
+
1
k
+
1
\int x^k\,dx=\frac{x^{k+1}}{k+1}
∫
x
k
d
x
=
k
+
1
x
k
+
1
.
Numerical
MEDIUM
3 marks
29 October 2023
24
Which of the following is true?
Comprehension
MEDIUM
2 marks
29 October 2023
25
Find
P
(
∣
X
∣
<
α
2
)
P(|X|<\frac{\alpha}{2})
P
(
∣
X
∣
<
2
α
)
. Enter the answer correct to two decimal places.
Comprehension
HARD
3 marks
29 October 2023
26
Let
X
X
X
be a continuous random variable with the following PDF:
f
X
(
x
)
=
k
(
1
+
x
)
2
f_X(x)=\frac{k}{(1+x)^2}
f
X
(
x
)
=
(
1
+
x
)
2
k
for
0
≤
x
≤
4
0\le x\le 4
0
≤
x
≤
4
, and
0
0
0
otherwise. Find the value of
k
k
k
. Enter the answer correct to two decimal places. Hint:
∫
1
(
a
+
b
x
)
2
d
x
=
−
1
b
(
a
+
b
x
)
\int\frac{1}{(a+bx)^2}\,dx=-\frac{1}{b(a+bx)}
∫
(
a
+
b
x
)
2
1
d
x
=
−
b
(
a
+
b
x
)
1
.
Numerical
MEDIUM
3 marks
16 July 2023
27
Find the value of
k
k
k
.
Comprehension
EASY
2 marks
16 July 2023
28
Find the value of
P
(
1
4
≤
X
≤
3
4
)
P(\frac{1}{4}\le X\le \frac{3}{4})
P
(
4
1
≤
X
≤
4
3
)
. Enter the answer correct to two decimal places.
Comprehension
MEDIUM
3 marks
16 July 2023
29
Which of the following statements are correct?
Single correct
MEDIUM
3 marks
16 October 2022
30
Among the options below, for what values of
a
a
a
and
b
b
b
is the function
f
f
f
a valid density function?
Comprehension
MEDIUM
3 marks
16 October 2022
31
With the choice of
a
a
a
and
b
b
b
given in the previous question, find
P
(
X
≤
−
1
2
∣
X
<
1
)
P\left(X\le -\frac{1}{2}\mid X<1\right)
P
(
X
≤
−
2
1
∣
X
<
1
)
. Enter the answer correct to three decimal places.
Comprehension
MEDIUM
2 marks
16 October 2022
32
Let
f
:
R
→
R
f:\mathbb{R}\to\mathbb{R}
f
:
R
→
R
be defined by
f
(
x
)
=
{
0
,
−
∞
<
x
<
0
,
1
4
,
0
<
x
≤
1
,
3
4
x
2
,
1
<
x
<
∞
.
f(x)=\begin{cases}0, & -\infty < x < 0,\\ \frac{1}{4}, & 0 < x \le 1,\\ \frac{3}{4x^2}, & 1 < x < \infty.\end{cases}
f
(
x
)
=
⎩
⎨
⎧
0
,
4
1
,
4
x
2
3
,
−
∞
<
x
<
0
,
0
<
x
≤
1
,
1
<
x
<
∞.
If a random variable
X
X
X
has the PDF
f
(
x
)
f(x)
f
(
x
)
, calculate
P
(
−
1
2
<
X
<
3
2
)
P\left(-\frac{1}{2}<X<\frac{3}{2}\right)
P
(
−
2
1
<
X
<
2
3
)
. Enter the answer correct to one decimal place. Hint: use
∫
a
b
1
x
2
d
x
=
1
a
−
1
b
\int_a^b \frac{1}{x^2}\,dx = \frac{1}{a}-\frac{1}{b}
∫
a
b
x
2
1
d
x
=
a
1
−
b
1
.
Numerical
MEDIUM
2 marks
05 June 2022
33
Find the value of
p
p
p
so that
f
X
(
x
)
f_X(x)
f
X
(
x
)
is a valid PDF. Enter the answer correct to two decimal places.
Numerical
EASY
2 marks
05 June 2022
34
Find the value of
P
(
X
<
4
∣
X
>
2
)
P(X<4\mid X>2)
P
(
X
<
4
∣
X
>
2
)
. Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
05 June 2022
35
What is the probability that Jadeja will bat for more than
40
40
40
minutes? Enter the answer correct to two decimal places.
Numerical
EASY
2 marks
05 June 2022
36
What is the probability that Jadeja will bat for more than
60
60
60
minutes given that he has already batted for more than
40
40
40
minutes? Enter the answer correct to two decimal places.
Numerical
MEDIUM
3 marks
05 June 2022
Showing 36 questions.
